A104244 -id:A104244 - cp's OEIS frontend

A344443 Completely additive with a(2)=5; for odd prime p, a(p) = ceiling((a(p-1) + a(p+1))/2).

0, 5, 8, 10, 12, 13, 14, 15, 16, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 24, 25, 25, 25, 25, 26, 26, 26, 26, 27, 27, 27, 27, 27, 27, 28, 28, 28, 28, 28, 28, 28, 29, 29, 29, 29, 29, 30, 29, 30, 30, 30, 30, 30, 30, 30, 30, 31, 31, 31, 31, 31, 31

Offset: 1

Peter Munn, May 19 2021

Monotonic until a(55) = 30 > 29 = a(56).
The length of a monotonic completely additive sequence using positive integers is limited by the size of the initial terms, and this constraint is related to the conflicts inherent in specifying musical scales. In either case, the route to a good compromise solution that avoids complexity can be seen as a search for rational approximations to ratios between logarithms of the first few prime numbers.
Here a(i)/a(j) is the intended approximation to log(i)/log(j), so by starting with a(2) = 5 we open the door to using 8/5 as an approximation to log(3)/log(2) = 1.58496... . This approximation underlies simple musical scales that divide an octave into 5, where a note of twice the frequency is 5 tones higher and a note of about 3 times the frequency is 8 tones higher. The most commonly used more complex musical scales divide an octave into 12, equivalent to starting a sequence like this with a(2) = 12 (see A344444).

			a(4) = a(2*2) = a(2) + a(2) from the definition of completely additive. So a(4) = 5 + 5 = 10.
3 is an odd prime number, so a(3) = ceiling((a(3-1) + a(3+1))/2). Using the values a(2) = 5 and a(4) = 10 that we already know, we get a(3) = ceiling((5 + 10)/2) = ceiling(7.5) = 8.
The sequence is defined as completely additive, so a(1) = 0, the identity element for addition. (To see this, note that "completely additive" implies a(2) = a(2*1) = a(2)+a(1), and solve the equation for a(1).)

Author?, Musical Temperaments and Ratios.
Encyclopedia of Mathematics, Additive arithmetic function.
Index entries for sequences related to music.

Equivalent sequence with a(2)=12, a(3)=19: A344444.
The first 11 terms match row 11 of A352957.
For other completely additive sequences see the references in A104244.
Cf. A020857, A060528, A119256.

a[1] = 0; a[n_] := a[n] = Plus @@ ((Last[#] * If[First[#] == 2, 5, Ceiling[(a[First[#] - 1] + a[First[#] + 1])/2]]) & /@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Jun 27 2021 *)

A344443(n) = if(1==n,0, my(f=factor(n)); sum(k=1,#f~,f[k,2]*if(2==f[k,1],5,ceil((1/2)*(A344443(f[k,1]-1)+A344443(f[k,1]+1)))))); \\ Antti Karttunen, May 19 2021

a(n*k) = a(n) + a(k).

A344444 Completely additive with a(2) = 12, a(3) = 19; for prime p > 3, a(p) = ceiling((a(p-1) + a(p+1))/2).

Original entry on oeis.org

0, 12, 19, 24, 28, 31, 34, 36, 38, 40, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 55, 56, 57, 57, 58, 59, 59, 60, 60, 61, 61, 62, 62, 63, 63, 64, 64, 65, 65, 66, 66, 66, 67, 67, 67, 68, 68, 68, 69, 69, 69, 70, 70, 70, 71, 71, 71, 72, 72, 72, 72, 73, 73, 73, 73, 74, 74

Offset: 1

Peter Munn, May 20 2021

Monotonic until a(143) = 87 > 86 = a(144).
The only infinite monotonic completely additive integer sequence is the all 0's sequence (cf. A000004). The challenge taken up here is to specify one that is monotonic for a modestly long number of terms, using a comparatively short prescriptive definition.
To start we specify values for a(2) and a(3) so that a(3)/a(2) approximates log(3)/log(2). 19/12 is a good approximation relative to the size of denominator. This reflects 2^19 = 524288 having a similar magnitude to 3^12 = 531441. Equivalently, we can say 3 is approximately the 19th power of the 12th root of 2. This approximation is used to construct musical scales. (See the Enevoldsen link, also A143800.) There is no better approximation with a denominator smaller than 29. [Revised by Peter Munn, Jun 14 2022]
To find a good specification to use for a(p) for larger primes, p, we are guided by knowing that if 2*a(n) < a(n-1) + a(n+1) then a completely additive sequence is not monotonic after a(n^2-1) because a(n^2) < a((n-1)*(n+1)) = a(n^2-1). Considering n = p, we see we want a(p) >= (a(p-1) + a(p+1))/2; but the same consideration for n = p-1 shows we don't want a(p) larger than necessary. These considerations lead towards the choice of "a(p) = ceiling((a(p-1) + a(p+1))/2)" for use in the definition.

			a(4) = a(2*2) = a(2) + a(2) from the definition of completely additive. So a(4) = 12 + 12 = 24. Similarly, a(6) = a(2*3) = a(2) + a(3) = 12 + 19 = 31.
5 is a prime number greater than 3, so a(5) = ceiling((a(5-1) + a(5+1))/2). Using the values a(4) = 24 and a(6) = 31 that we calculated earlier, we get a(5) = ceiling((24 + 31)/2) = ceiling(27.5) = 28.
The sequence is defined as completely additive, so a(1) = 0, the identity element for addition. (To see this, note that "completely additive" implies a(2) = a(2*1) = a(2)+a(1), and solve the equation for a(1).)

Encyclopedia of Mathematics, Additive arithmetic function
Keith Enevoldsen, Twelve-Tone Musical Scale
Index entries for sequences related to music

Equivalent sequence with a(2)=5, a(3)=8: A344443.
First 10 terms match A143800.
Cf. row 23 of A352957.
Cf. A000004, A020857, A060528, A061920.
For other completely additive sequences see the references in A104244.

a[1] = 0; a[n_] := a[n] = Plus @@ ((Last[#] * Which[First[#] == 2, 12, First[#] == 3, 19, First[#] > 3, Ceiling[(a[First[#] - 1] + a[First[#] + 1])/2]]) & /@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Jun 27 2021 *)

a(n*k) = a(n) + a(k).

A277625 Nontrivial values of Fibonacci polynomials.

Original entry on oeis.org

2, 3, 5, 8, 10, 12, 13, 17, 21, 26, 29, 33, 34, 37, 50, 55, 65, 70, 72, 82, 89, 101, 109, 122, 135, 144, 145, 169, 170, 197, 226, 228, 233, 257, 290, 305, 325, 357, 360, 362, 377, 401, 408, 442, 485, 528, 530, 577, 610, 626, 677, 701, 730, 747, 785, 842, 901, 962, 985, 987

Offset: 1

Bobby Jacobs, Oct 24 2016

nonn

The polynomial FibonacciPolynomial(x, y) satisfies the recurrence FibonacciPolynomial(0, y) = 0, FibonacciPolynomial(1, y) = 1, and FibonacciPolynomial(x, y) = y*FibonacciPolynomial(x-1, y) + FibonacciPolynomial(x-2, y).
Nontrivial means a value FibonacciPolynomial(x, y) with x>=3 and y>=1. For FibonacciPolynomial(0, y) = 0 and FibonacciPolynomial(1, y) = 1 for all y, and any number y can be represented trivially as FibonacciPolynomial(2, y).
5 = FibonacciPolynomial(5, 1) = FibonacciPolynomial(3, 2) is the only known number that can be represented as a nontrivial Fibonacci polynomial in more than one way.
Numbers obtained as A104244(n,A206296(k)), where n >= 1 and k >= 3 (all terms from array A073133 except its two leftmost columns) and then sorted into ascending order, with any possible duplicate (5) removed. - Antti Karttunen, Oct 29 2016

			12 is in this sequence because FibonacciPolynomial(4, 2) = 12.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Wikipedia, Fibonacci polynomials.

Cf. A000045, A000129, A001076, A006190, A052918 (FibonacciPolynomial(x, y) for different values of y).
Cf. A002522, A054602, A085151 (FibonacciPolynomial(x, y) for different values of x).
Cf. A073133, A104244, A206296.

Take[Union[Flatten[Table[Fibonacci[x, y], {x, 3, 20}, {y, 50}]]], 60] (* Robert G. Wilson v, Oct 24 2016 *)

list(lim)=my(v=List()); for(y=1,sqrtint(lim\1-1), my(a=y,b=y^2+1); while(b<=lim, listput(v,b); [a,b]=[b,a+y*b])); Set(v) \\ Charles R Greathouse IV, Oct 30 2016

FibonacciPolynomial(x, y) with x>=3 and y>=1.

a(n) = n^2 - 2*n^(5/3) - O(n^(3/2)). - Charles R Greathouse IV, Nov 03 2016

More terms from Robert G. Wilson v, Oct 24 2016

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A344443 Completely additive with a(2)=5; for odd prime p, a(p) = ceiling((a(p-1) + a(p+1))/2).

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A344444 Completely additive with a(2) = 12, a(3) = 19; for prime p > 3, a(p) = ceiling((a(p-1) + a(p+1))/2).

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A277625 Nontrivial values of Fibonacci polynomials.

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